Problem: Evaluate $\int\sec x\,dx\,$. Choose 1 answer: Choose 1 answer: (Choice A) A $\sec x\tan x+C$ (Choice B) B $\sec x+\tan x+C$ (Choice C) C $\dfrac{\sec x}{\ln|\sec x+\tan x|}+C$ (Choice D) D $\ln|\sec x+\tan x|+C$
Answer: This is a problem that you basically have to MEMORIZE the answer to, since learning how to do it is equivalent to memorizing the answer!! [ And the only ones who really have to learn how to do this are your teachers! ] Multiply the numerator and the denominator of the integrand by $~\sec x+\tan x\,$. $\begin{aligned} &\phantom{=}\int\sec x\,dx \\\\ &=\int\sec x\dfrac{\sec x+\tan x}{\sec x+\tan x}\,dx \\\\ &= \int \dfrac{\sec^2x+\sec x\tan x}{\sec x+\tan x}\,dx \end{aligned}$ Now, miraculously, the numerator is the derivative of the denominator!! $\int \dfrac{\sec^2x+\sec x\tan x}{\sec x+\tan x}\,dx= \ln|\sec x+\tan x|+C$